# Poincaré recurrence theorem

Let $(X,\mathscr{S},\mu)$ be a probability space and let $f\colon X\to X$ be a measure preserving transformation.

###### Theorem 1.

For any $E\in\mathscr{S}$, the set of those points $x$ of $E$ such that $f^{n}(x)\notin E$ for all $n>0$ has zero measure. That is, almost every point of $E$ returns to $E$. In fact, almost every point returns infinitely often; i.e.

 $\mu\left(\{x\in E:\textnormal{ there exists }N\textnormal{ such that }f^{n}(x)\notin E\textnormal{ for all }n>N\}\right)=0.$

The following is a topological version of this theorem:

###### Theorem 2.

If $X$ is a second countable Hausdorff space and $\mathscr{S}$ contains the Borel sigma-algebra, then the set of recurrent points of $f$ has full measure. That is, almost every point is recurrent.

Title Poincaré recurrence theorem PoincareRecurrenceTheorem 2013-03-22 14:29:50 2013-03-22 14:29:50 Koro (127) Koro (127) 6 Koro (127) Theorem msc 37B20 msc 37A05